# Numeric Functional Regression coefficients

I'm working on some data where I have thousands of curves in the defined as: $P(t)$ that basically represent eluition profiles of some proteins.

My aim is to represent each of this $P(t)$ curves as a linear combination of some other curves plus some error. More specifically:

$P(t) = \beta_{1} M(t) + \beta_2L(t)+\beta_3Z(t)+\beta_4G(t)+\epsilon$

Basically I want to find the scalar coefficient $\beta_1,\beta_2,..,\beta_n$ that minimize the residual sum of squares on $P(t)$ for me it seems like a simple multiple regression problem. The only issue seems to be that we are talking about functional data.

It seems that maybe can be a very simple problem, but I don't really know if I can just use simple linear regression to estimate the coefficients. I've heard about functional linear regression that takes the forms:

$P(t) = \beta_0(t)+\beta_1(t)M(t)+\beta_2(t)L(t)+\epsilon$

But in this case the coefficients are defined as functions, I really need them to be expressed as a scalar. Maybe I'm just a little bit confused, but It seems a very simple problem, there is a feasible way to resolve it?

• Do you have a preferred programming language? Jul 14, 2018 at 12:43
• Yes, I use mainly R. It really seems a simple problem, but I'm not sure if I can resolve it with the classical lm() function. Unfortunately I think no. Jul 14, 2018 at 13:31
• Any hints or suggestions? :) Jul 15, 2018 at 12:04
• For y = f(x): if I have three vectors where the first vector is all 1.0, the second contains the values of x, and the third contains the values of x^2, when I perform linear regression with linear algebra of y against these vectors I will have the coefficients a, b, c for the equation "y = a + bx + cx^2". Might you do something similar in this case, where you create vectors of the function values and then use standard linear algebra? Jul 15, 2018 at 13:04